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Properties

Label 20160bc
Number of curves $4$
Conductor $20160$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -26463, -6898988]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -26463, -6898988]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -26463, -6898988]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 20160bc have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 20160bc do not have complex multiplication.

Modular form 20160.2.a.bc

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{5} - q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 20160bc

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20160.bg4 20160bc1 \([0, 0, 0, -26463, -6898988]\) \(-43927191786304/415283203125\) \(-19375453125000000\) \([2]\) \(122880\) \(1.8076\) \(\Gamma_0(N)\)-optimal
20160.bg3 20160bc2 \([0, 0, 0, -729588, -239211488]\) \(14383655824793536/45209390625\) \(134994517056000000\) \([2, 2]\) \(245760\) \(2.1542\)  
20160.bg1 20160bc3 \([0, 0, 0, -11664588, -15333885488]\) \(7347751505995469192/72930375\) \(1742151462912000\) \([2]\) \(491520\) \(2.5008\)  
20160.bg2 20160bc4 \([0, 0, 0, -1044588, -12537488]\) \(5276930158229192/3050936350875\) \(72880377029849088000\) \([2]\) \(491520\) \(2.5008\)